Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 06:
Semi-Analytical Rate Relations
for Oil and Gas Flow
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 1
Rate Relations for Oil and Gas Flow
Historical Perspectives
 "Backpressure" equation.
 Arps relations (exponential, hyperbolic, and harmonic).
 Derivation of Arps' exponential decline relation.
 Validation of Arps' hyperbolic decline relation.
Specialized Gas Flow Relations:
 Fetkovich Gas Flow Relation.
 Ansah-Buba-Knowles Gas Flow Relations.
Specialized Oil Flow Relations:
 Fetkovich Oil Flow Relation.
Inflow Performance Relations (IPR):
 Early work (for rationale).
 Oil IPR and Solution-Gas Drive IPR.
 Gas Condensate IPR.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 2
History: Deliverability/"Backpressure" Equation
Gas Well Deliverability:
 The original well deliverability
relation was completely empirical (derived from observations),
and is given as:
qg  C( p2  p2 )n
wf
 This relationship is rigorous for
low pressure gas reservoirs,
(n=1 for laminar flow).
 From: Back-Pressure Data on NaturalGas Wells and Their Application to
Production Practices — Rawlins and
Schellhardt (USBM Monograph, 1935).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 3
History: p2 Diffusivity Equations
 Diffusivity Equations for a "Dry Gas:" p2 Relations
 p2 Form — Full Formulation:
2
2
 (p )

p
[ln(  g z )]( p ) 
2
2 2
 g ct 
k
t
( p2 )
 p2 Form — Approximation:
2
2
 (p ) 
PETE 613
(2005A)
 g ct 
k
t
( p2 )
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 4
History: Gas p2 Condition (gz vs. p, T=200 Deg F)
 "Dry Gas" PVT Properties: (gz vs. p)
 Basis for the "pressure-squared" approximation (i.e., use of p2 variable).
 Concept: (gz) = constant, valid only for p<2000 psia.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 5
History: Gas p2 Condition (gz vs. p, T=200 Deg F)
 g z 
p
p
ppg  
dp

p
p

z

 pn base g

 "Dry Gas" PVT Properties: (gz vs. p)
 Concept: IF (gz) = constant, THEN p2-variable valid.
 (gz)  constant for p<2000 psia.
 Even with numerical solutions, p2 formulation would not be appropriate.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 6
History: "Arps" Equations
Arps' (Empirical) Rate Relations:
 Exponential decline case (conservative).
 Harmonic decline case (liberal).
 Hyperbolic decline case (everything in between).
Fetkovich (Radial Flow) Decline Type Curve:
 Exponential, hyperbolic, harmonic decline cases.
Derivation of the Arps' Exponential Rate Relation:
 Combination of liquid material balance and liquid pseudosteady-state flow equation solved for pwf  constant.
 Useful for deriving auxiliary relations (cumulative production
functions, in particular).
Derivation of the Arps' Hyperbolic Rate Relation:
 Interesting exercise, limited practical value.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 7
Arps Relations: Summary
Flowrate-Time Relations:
Exponential: (b=0)
Hyperbolic: (0<b<1)
Harmonic: (b=1)
(1/2)
q  qi exp(  Di t )
q
qi
(1 bDi t )1 / b
qi
q
(1  Di t )
Cumulative Production-Time Relations:
Exponential: (b=0)
Hyperbolic: (0<b<1)
Harmonic: (b=1)
PETE 613
(2005A)
q
N p  i [1  exp(  Di t )]
Di
qi
Np 
[1  (1  bDi t )11 / b ]
(1  b) Di
q
N p  i ln(1  Di t )
Di
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 8
Arps Relations: Summary
(2/2)
Flowrate-Cumulative Production Relations:
Exponential: (b=0)
q  qi  Di N p
Plot of: q versus Np
Hyperbolic: (0<b<1)
q1b 
(1  b) Di
qi
b
(N  N p )
qi b
or ( N  N p ) 
q1b Plot of: log(N-Np) versus log(q)
(1  b) Di
Harmonic: (b=1)
 D

q  qi exp  i N p 
 qi

PETE 613
(2005A)
Plot of: log(q) versus Np
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 9
Arps Relations: Example 1
Sewell Ranch Well No. 1 — Barnett Field (NorthTexas)
2000
1800
Gas Flowrate
Exponential Rate Model
Hyperbolic Rate Model
Wellbore Pressure
1600
1.E+03
1400
1200
1000
1.E+02
800
600
1.E+01
400
Flowing Tubing Pressure, psig
Gas Production Rate, MSCFD
1.E+04
200
1.E+00
0
500
1000
1500
2000
2500
3000
3500
0
4000
Producing Time, days
a. Semilog "Rate-Time" Plot: Barnett Gas
Field.
(1/2)
a. q  qi exp(  Di t )
qi
q
(1 bDi t )1 / b
(Exponential)
(Hyperbolic)
b. q  qi  Di N p
(Exponential)
1
1b
 qi
q

 (1  b) Di 
b
1
( N  N p )1b
(Hyperbolic)
qi b
c. ( N  N p ) 
q1b
(1  b) Di
(Hyperbolic)
Sewell Ranch Well No. 1 — Barnett Field (NorthTexas)
1.E+07
Sewell Ranch Well No. 1 — Barnett Field (NorthTexas)
(G-Gp) Data Function
Hyperbolic Model
1400
Cumulative Gas Production
Exponential Model
1200
Hyperbolic Model
1.E+06
(G-Gp), MSCF
Gas Production Rate, MSCFD
Exponential Model
1000
800
600
1.E+05
400
200
Method is designed for hyperbolic decline case
0
0
250,000
500,000
750,000
1,000,000 1,250,000 1,500,000
1.E+04
1.E+01
Cumulative Gas Production, MSCF
b. Cartesian "Rate-Cumulative" Plot:
Barnett Gas Field (North Texas).
PETE 613
(2005A)
1.E+02
1.E+03
1.E+04
Gas Production Rate, MSCFD
c. Log-Log "(G-Gp)-Rate" Plot: Barnett Gas
Field (North Texas).
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 10
Arps Relations: Example 1
(2/2)
Sewell Ranch Well No. 1 — Barnett Field (NorthTexas)
2000
Gas Flowrate
Exponential Rate Model
Hyperbolic Rate Model
Wellbore Pressure
1800
1600
1.E+03
1400
1200
1000
1.E+02
800
600
1.E+01
400
Flowing Tubing Pressure, psig
Gas Production Rate, MSCFD
1.E+04
200
1.E+00
0
500
1000
1500
2000
2500
3000
3500
0
4000
Producing Time, days
q  qi exp(  Di t )
(Exponential)
q
qi
(1 bDi t )1/ b
(Hyperbolic)
 EUR Analysis: Barnett Field (North Texas (USA))
 Semilog "Rate-Time" Plot: Barnett Gas Field.
 Note data scatter and apparent fit of hyperbolic function.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 11
Arps Relations: Example 2
SPE 84287 — East TX Gas Well 1 (Low Permeability Gas)
3000
Gas Flowrate
Exponential Rate Model
Hyperbolic Rate Model
Wellbore Pressure
2750
2500
2250
1.E+04
2000
1750
1500
1250
1.E+03
1000
750
500
Flowing Tubing Pressure, psig
Gas Production Rate, MSCFD
1.E+05
250
1.E+02
0
50
100
150
200
250
300
0
350
Producing Time, days
a. Semilog "Rate-Time" Plot: SPE 84287 —
East Texas Gas Well 1.
(1/2)
a. q  qi exp(  Di t )
qi
q
(1 bDi t )1 / b
(Exponential)
(Hyperbolic)
b. q  qi  Di N p
(Exponential)
1
1b
 qi
q

 (1  b) Di 
b
1
( N  N p )1b
(Hyperbolic)
qi b
c. ( N  N p ) 
q1b
(1  b) Di
(Hyperbolic)
SPE 84287 — East TX Gas Well 1 (Low Permeability Gas)
1.E+07
SPE 84287 — East TX Gas Well 1 (Low Permeability Gas)
(G-Gp) Data Function
Exponential Model
Hyperbolic Model
9000
Cumulative Gas Production
8000
Exponential Model
7000
Hyperbolic Model
1.E+06
(G-Gp), MSCF
Gas Production Rate, MSCFD
10000
6000
5000
4000
3000
1.E+05
2000
1000
0
Method is designed for hyperbolic decline case
0
250,000
500,000
750,000
1,000,000 1,250,000 1,500,000
Cumulative Gas Production, MSCF
1.E+04
1.E+01
1.E+02
1.E+03
1.E+04
Gas Production Rate, MSCFD
b. Cartesian "Rate-Cumulative" Plot: SPE
84287 — East Texas Gas Well 1.
PETE 613
(2005A)
c. Log-Log "(G-Gp)-Rate" Plot: SPE 84287 —
East Texas Gas Well 1.
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 12
Arps Relations: Example 2
(2/2)
SPE 84287 — East TX Gas Well 1 (Low Permeability Gas)
1.E+05
6000
5000
4500
1.E+04
4000
3500
3000
2500
1.E+03
2000
1500
Flowing Tubing Pressure,
psig
Gas Production Rate, MSCFD
5500
Gas Flowrate
Exponential Rate Model
Hyperbolic Rate Model
Wellbore Pressure
1000
500
1.E+02
0
50
100
150
200
250
300
0
350
Producing Time, days
q  qi exp(  Di t )
(Exponential)
q
qi
(1 bDi t )1/ b
(Hyperbolic)
 EUR Analysis: SPE 84278 Well 1 (East Texas (USA))
 Combination "Rate-Time" and "Pressure-Time" plot.
 Note pressure buildup (used to check with PTA).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 13
Fetkovich Decline Type Curve: Empirical
 Fetkovich "Empirical" Decline Type Curve:
 Log-log "type curve" for the Arps "decline curves" (Fetkovich, 1973).
 Initially designed as a graphical solution of the Arps' relations.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 14
Analytical Type Curves: Radial Flow
 "Analytical" Rate Decline Curves:
 From: SPE 04629 — Fetkovich (1973).
 Data from van Everdingen and
Hurst (1949), replotted as a rate
decline plot (Fetkovich, 1973).
 This looks promising — but this is
going to be one really big "type
curve."
 What can we do? Try to collapse
all of the trends to a single trend
during boundary-dominated flow
(Fetkovich, 1973).
 "Analytical" stems are another
name for transient flow behavior,
which can yield estimates of
reservoir flow properties.
 From: SPE 04629 — Fetkovich (1973).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 15
Fetkovich Decline Type Curve: Analytical
 Fetkovich "Analytical" Decline Type Curve: (constant pwf)
 Log-log "type curve" for transient flow behavior (Fetkovich, 1973).
 First "tie" between pressure transient and production data analysis.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 16
Fetkovich Decline Type Curve: Composite
 Fetkovich "Composite" Decline Type Curve:
 Assumes constant bottomhole pressure production.
 Radial flow in a finite radial reservoir system (single well).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 17
Derivation: Arps' Exponential Decline Case
Oil Material Balance Relation:
1 Bo
p  pi 
Np
Nct Boi
Oil Pseudosteady-State Flow Relation:
o Bo  1  4 1 A  
p  pwf  bo, pss qo bo, pss  141.2
 ln  
  s
2
kh  2  e C A rw  
Steps:
1. Differentiate both relations with respect to time.
2. Assume pwf = constant (eliminates d(pwf)/dt term).
3. Equate results, yields 1st order ordinary differential equation.
4. Integrate.
5. Exponentiate result.

1
q  qi exp Di t 
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
1 Bo 
 Di 

b
Nc
B

o, pss
t oi 
Slide — 18
Validation: Arps' Hyperbolic Decline Case
(Details of derivation are omitted, see paper
SPE 19009, Camacho and Raghavan (1989)).
a. Hyperbolic flowrate relations for the case
of constant pressure production from a
solution gas drive reservoir (Camacho
and Raghavan (1989)).
PETE 613
(2005A)
b. Hyperbolic decline type curve with data
simulation performance data superimposed (Camacho and Raghavan (1989)).
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 19
Specialized Gas Flow Relations
 Fetkovich Gas Flow Relation (poor approximation):
 Rate-time.
 Characteristic behavior plot.
 Results from Knowles-Ansah-Buba work:
 Rate-time.
 Rate-cumulative.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 20
Fetkovich Gas Flow Relation: Poor Approximation
Gas Material Balance Relation: (z=1 ! (ideal gas?))
p
p  pi  i Gp ( z  zi  1)
G
Gas Pseudosteady-State Flow Relation: (Fetkovich)
2
2 n
q g  C g ( p  pwf
)
Final Result: (Fetkovich)
qg
q gi

PETE 613
(2005A)
1
2n

 q gi   2n  1
1  (2n  1) 
t 
 G  

( pwf  0)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 21
Fetkovich Decline Type Curve: Gas
 Fetkovich "Analytical" Gas Decline Type Curve: (pwf = 0)
 Cheated (z=1) ... this is not a valid solution (Fetkovich, 1973).
 Good intentions ... wanted to develop a "simple" gas solution.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 22
Knowles — Gas Rate-Time Relation
 "Knowles" rate-time relation for gas flow:
 Models decline of gas flowrate versus time.
 Better representation of rate-time behavior than the "Arps"
hyperbolic decline relations.
  1  p 


wD
 1  

 exp(  pwD t Dd ) 
   1  pwD 
  1



)    1  pwD 

 exp(  pwD t Dd ) 
1  
   1  pwD 




2
q gDd
qg
2
pwD


q gi (1  pwD2
t Dd 
2qgi
  pwf / z wf  
1  
G

  pi / zi  
pwD
2
t
 pwf / z wf 


p
/
z
 i i 
Assumptions:
 Volumetric, dry gas reservoir.
 pi < 6000 psia.
 Constant bottomhole flowing pressure.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 23
Knowles-Buba — Gas Rate-Cumulative Relation
This work presents an analysis and interpretation sequence for the estimation of reserves in a volumetric
dry-gas reservoir. This is based on the "Knowles" ratecumulative production relation for pseudosteady-state
gas flow given as:
q g  q gi 
2qgi
2
 p /z 
1   wf wf   G
  pi / zi  


Gp 
qgi
 p /z
1   wf wf
  pi / zi




2
2
Gp
 G2


"Knowles" relations for gas flow:
 qg — Gp follows quadratic "rate-cumulative" relation.
 Approximation valid for pi<6000 psia.
 Assumes pwf = constant.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 24
Simplified Gas Flow: Validation of Knowles Eqs.
b. Simulated Performance Case: Gp versus t
(pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF).
 qg vs. t and Gp vs. t:
a. Simulated Performance Case: qg versus t
(pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF).
PETE 613
(2005A)
 Base plots ― verify models by
Ansah, et. al
 Comparative trends of 0.9qgi , qgi
and 1.1qgi . Comparison applied
to all analysis plots.
 Very good match on both plots,
accuracy verifies model.
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 25
Simplified Gas Flow: Validation of Buba Eq.
qg  qgi  DiGp 
Di 
1 Di 2
Gp
2 G
2qgi
 p /z
1   wf wf
  pi / zi




2
G


"Knowles-Buba" relations for gas flow:
 Simulated performance case: qg-Gp (quadratic "rate-cumulative").
 pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF.
 Data function matches well with quadratic model function.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 26
Specialized Oil Flow Relations
 Fetkovich Oil Flow Relation:
 Rate-time (Decline Type Curve Analysis).
 Deliverability (Isochronal Testing of Oil Wells).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 27
Fetkovich Oil Flow Relation: (Approximation)
Oil Material Balance Relation: (p2 – formulation!)
( pi ) 2
( p )  ( pi ) 
Np
N
Oil Pseudosteady-State Flow Relation: (Fetkovich)
 p 2
2 n
qo  J oi   ( p  pwf
)
 pi 
Final Result: (Fetkovich)
2
2
qo
1

qoi  1  q   2n  1
oi t
1

 2 N  



PETE 613
(2005A)
( pwf  0)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 28
Fetkovich Decline Type Curve: Solution Gas Drive
 Fetkovich "Analytical" Oil Decline Type Curve: (pwf = 0)
 Cheated (pressure-squared material balance relation?) ... this is not a
valid solution (Fetkovich, 1973).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 29
Oil "Backpressure" Relation: Fetkovich
a. Deliverability ("backpressure") plot
developed for Well 2/4-2X prior to matrix
acidizing treatment. (Fetkovich [SPE
004529 (1973)]).
PETE 613
(2005A)
(1/2)
b. Deliverability ("backpressure") plot
developed for Well 2/4-2X after matrix
acidizing treatment. Note much higher
flowrate performance and apparent nonlinear (i.e., non-laminar) flow behavior
(Fetkovich [SPE 004529 (1973)]).
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 30
Oil "Backpressure" Relation: Fetkovich
a. Comparison of simulated and predicted
IPR behaviors for solution-gas-drive case
(Vogel [SPE 001476 (1968)]).
PETE 613
(2005A)
(2/2)
b. Deliverability ("backpressure") plot
developed using Vogel data. Proof of
concept for "backpressure" flow relation
(Fetkovich [SPE 004529 (1973)]).
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 31
Inflow Performance Relations (IPR)
 Early work (for rationale)
 Oil IPR and Solution-Gas Drive IPR
 Vogel IPR work (for familiarity with approach)
 Other IPR work (for reference/orientation)
 Gas Condensate IPR
 Fevang and Whitson work (for reference)
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 32
History Lessons — Early Performance Relations
Early “Gas Deliverability Plot,"
note the straight-line trends for
the data (circa 1935).
Early “Gas IPR Plot," note
the quadratic relationship
between wellhead pressure
and flowrate (circa 1935).
 Well deliverability analysis: (after Rawlins and Schellhardt)
 These plots represent the earliest attempts to quantify behavior and to
predict future performance.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 33
History Lessons — "Backpressure" Equation
Gas Well Deliverability:
 The original well deliverability
relation was derived from
observations:
qg  C( p2  p2 )n
wf
 The "inflow performance relationship" (or IPR) for this case is:
(assuming n=1)
qg  C ( p 2  p 2 )
wf
2
 From: Back-Pressure Data on NaturalGas Wells and Their Application to
Production Practices — Rawlins and
Schellhardt (USBM Monograph, 1935).
PETE 613
(2005A)
qg ,max  C( p ) ( p  0)
wf
 p
2


qg
 wf 

 1 



q g ,max
p




Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 34
History Lessons — IPR Developments/Correlations
 p

p
2


qo
wf 
wf 


qo,max 1  0.2  p   0.8  p 




p

qg

wf 

qg,max 1   p 

qo
qo,max 1 
Early "Inflow Plot," an attempt
to correlate well rate and pressure behavior — and to establish the maximum flowrate,
(after Gilbert (1954)).






pwf
p
2







IPR "comparison" — liquid (oil),
gas, and "two-phase" (solution
gas-drive) cases presented to
illustrate comparative behavior
(after Vogel (1968)).
 Inflow Performance Relationship (IPR):
 Correlate performance, estimate maximum flowrate.
 Individual phases require, separate correlations.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 35
Solution-Gas Drive Systems — Vogel IPR
Vogel Correlation: (Statistical)

p
2
p
qo 1  0.2 wf   0.8  wf 


qo,max


p 
p








IPR behavior is dependent on
the depletion stage (i.e., the
level of reservoir depletion).
No single correlation of IPR
behavior is possible.
The Vogel IPR correlation and
its variations are well established as the primary performance
prediction relations for production engineering applications.
The original correlation is derived from reservoir simulation.
 Vogel IPR Correlation: Solution Gas-Drive Behavior
 Derived as a statistical correlation from simulation cases.
 No "theoretical" basis — Intuitive correlation (qo,max and pavg).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 36
Solution-Gas Drive Systems — Other Approaches
Fetkovich IPR: (Semi-Empirical)









p
qo  1  wf
qo,max
p














2 
n







Richardson, et al. IPR: (Empirical)

p
2
p


qx 1  ν wf   (1  ν )  wf 
x
x 

qx,max


p 
p








(x = phase (e.g., oil, gas, water))
 Other IPR Correlations:
 Fetkovich: Derived assuming linear mobility-pressure relationship.
 Richardson, et al.: Empirical, generalized correlation.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 37
Solution-Gas Drive Systems— Other Approaches
Wiggins, et al. IPR: (Semi-Rigorous)

p
2
p
3
p




qo 1  a wf   a  wf   a  wf   ...
1 p 
2  p 
3  p 
qo,max











 Other IPR Correlations:
 Wiggins, et al.: Used a polynomial expansion of the mobility function in
order to yield a semi-rigorous IPR formulation.
 Coefficients (a1, a2…) are determined based on the mobility function
and its derivatives taken at the average reservoir pressure.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 38
Solution-Gas Drive Systems— Other Approaches
Pseudopressure Formulation – Oil Phase



p

 k

μ
B
o

 dp
p po( p)  o o 


ko  p p
 μo Bo 

n base 







Mobility Function






ko   f ( p)  a  2bp
μoBo  p

p
2
p
qo 1  ν wf   (1  ν )  wf 
o
o 

qo,max


p 
p








 Other IPR Correlations:
 n strong function of pressure and saturation.
 Semi-rigorous IPR formulation (derived for the solution-gas case) has
the same form of the Richardson, et al. IPR (which is empirical).
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 39
Gas Condensate Systems — Pseudopressure
Three flow regions were
characterized:
 Region 1 — Main cause of
productivity loss, oil and gas flow
simultaneously.
 Region 2 — Two phases coexist,
but only gas is mobile.
 Region 3 — single-phase gas.

p  k

k
1
k
h
o
o

 dp
qg 

R
s



μo Bo 
141.2 ln(re /rw) 3/4  s p  μo Bo


wf

 Fevang and Whitson Correlation: Gas Condensate systems
 Pressure and saturation functions need to be know in advance —
GOR, PVT properties and relative permeabilities.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 40
Gas Condensate IPR — Del Castillo 2003 (TAMU)
Model-Based Performance Study:
 Radial, fully compositional, single well simulation model
 Parameters/functions used in simulation:
 Reservoir Temperature: T = 230, 260, 300 Deg F
 Critical Oil Saturation: Soc = 0, 0.1, 0.3
 Residual Gas Saturation: Sgr = 0, 0.15, 0.5
 Relative Permeability: 7 sets of kro-krg data
 Fluid Samples: 4 synthetic cases, 2 field samples
 Assumptions used in simulation:
 Interfacial tension effects are neglected
 Non-Darcy flow effects are neglected
 Capillary pressure effects are neglected
 Refined simulation grid in the near-well region
 Skin effect is neglected
 Gravity and composition gradients are neglected
 Simulations begun at the dew point pressure
 Correlation of gas and gas-condensate performance using Richardson
IPR model.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 41
Gas Condensate — IPR Trends (Condensate)
IPR Curves - Condensate Production
(Case16)
6000
0.8
Np/N = 0.18%
Np/N = 0.36%
Np/N = 1.79%
Np/N = 3.58%
Np/N = 5.37%
Np/N = 7.15%
Np/N = 8.94%
Np/N = 10.73%
4000
3000
2000
0.7
qo/qo,max
pwf , psia
1
0.9
Legend
5000
Normalized Oil Flowrate
(Case16)
0.6
Legend
0.5
Np/N = 0.18%
Np/N = 0.36%
Np/N = 1.79%
Np/N = 3.58%
Np/N = 5.37%
Np/N = 7.15%
Np/N = 8.94%
Np/N = 10.73%
IPR Model
0.4
0.3
0.2
1000
0.1
0
0
0
200
400
600
800
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
p wf /p bar
Normalized Oil Flowrate
Dimensionless IPR plot
(condensate) — Case 16 (gas
condensate system)
Base IPR plot (condensate)
—
(Case16)
1
Case 16
0.9 (gas condensate system). 0.8
0.7
qo/qo,max
0
q o , STB/D
0.6
0.5
 Condensate
IPR Correlations (gas condensate reservoirs)
0.4
Np/N = 0.18%
Np/N = 0.36%
Np/N = 1.79%
Np/N = 3.58%
Np/N = 5.37%
Np/N = 7.15%
Np/N = 8.94%
Np/N = 10.73%
IPR Model
 All eight
depletion stages regressed simultaneously.
0.3
0.2
 Excellent
correlation — all stages.
0.1
PETE 613
(2005A)
0
Semi-Analytical Rate Relations
p wf /p bar
for Oil and Gas Flow
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Slide — 42
Gas Condensate — IPR Trends (Gas)
IPR Curves - Gas Production
(Case16)
6000
0.9
0.8
0.7
Gp/G = 0.09%
Gp/G = 0.47%
Gp/G = 0.95%
Gp/G = 4.75%
Gp/G = 9.5%
Gp/G = 23.74%
Gp/G = 47.48%
Gp/G = 66.48%
4000
3000
2000
qg/qg,max
pwf , psia
1
Legend
5000
Normalized Gas Flowrate
(Case16)
0.6
Legend
0.5
Gp/G = 0.09%
Gp/G = 0.47%
Gp/G = 0.95%
Gp/G = 4.75%
Gp/G = 9.5%
Gp/G = 23.74%
Gp/G = 47.48%
Gp/G = 66.48%
IPR Model
0.4
0.3
0.2
1000
0.1
0
0
0
1000
2000
3000
4000
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
p wf /p bar
q g , MSCF/D
Normalized Gas Flowrate
Base IPR plot
(gas)
— Case
(Case16)
1
16 (gas0.9 condensate system).
0.8
1
Dimensionless IPR plot (gas)
— Case 16 (gas condensate
system).
qg/qg,max
0.7
0.6
0.5
 Gas IPR
Correlations (gas condensate reservoirs)
Gp/G = 0.09%
Gp/G = 0.47%
Gp/G = 0.95%
Gp/G = 4.75%
Gp/G = 9.5%
Gp/G = 23.74%
Gp/G = 47.48%
Gp/G = 66.48%
IPR Model
0.4
 All eight
depletion stages regressed simultaneously.
0.3
 Excellent
correlation — even when there is a more pronounced curve
0.2
0.1
overlap
(gas).
0
PETE 613
(2005A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p wf /p bar
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 43
Gas Condensate — Difference in IPR Trends
IPR Curves - Condensate Production
(Case16)
6000
6000
Legend
5000
Np/N = 0.18%
Np/N = 0.36%
Np/N = 1.79%
Np/N = 3.58%
Np/N = 5.37%
Np/N = 7.15%
Np/N = 8.94%
Np/N = 10.73%
4000
3000
2000
Legend
Np/N = 0.43%
Np/N = 0.86%
Np/N = 4.29%
Np/N = 8.59%
Np/N = 12.88%
Np/N = 17.17%
Np/N = 21.46%
Np/N = 25.76%
5000
pwf , psia
pwf , psia
IPR Curves - Condensate Production
(Case1)
1000
4000
3000
2000
1000
0
0
0
200
400
600
800
0
100
200
q o , STB/D
Oil Flowrate
Base IPR Normalized
plot (condensate)
—
(Case16)
1
Case 16
(Very rich gas
0.9
condensate
system).
0.8
0.7
0.6
Np/N = 0.18%
Np/N = 0.36%
Np/N = 1.79%
Np/N = 3.58%
Np/N = 5.37%
Np/N = 7.15%
Np/N = 8.94%
Np/N = 10.73%
IPR Model
qo/qo,max
qo/qo,max
400
Normalized
Oil Flowrate
Base IPR plot
(condensate)
—
(Case1)
Case 1 1(Lean gas condensate
0.9
system).
0.8
0.7
0.5
300
q o , STB/D
0.6
0.5
Np/N = 0.43%
Np/N = 0.86%
Np/N = 4.29%
Np/N = 8.59%
Np/N = 12.88%
Np/N = 17.17%
Np/N = 21.46%
Np/N = 25.76%
IPR Model
0.4
0.4
 Condensate
IPR Shape (gas condensate reservoirs)
0.3
0.3
 Remarkable
difference in shape between a 0.2
very rich gas condensate
0.2
system
and a lean one.
0.1
0.1
0
PETE 613
(2005A)
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p wf /p bar Semi-Analytical Rate Relations
for Oil and Gas Flow
p wf /p bar
Slide — 44
0.2
Gas Condensate — IPR Parameter (no or ng )
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p wf /p bar
Dimensionless IPR curves
Dimensional IPR curves
1
0.9
6000
0.8
pwf , psia
0.7
qo,g/qo,g,max
Legend
5000
0.6
0.5
Legend
Legend = 0.15
0.4
no,g
no,g
no,g
no,g
no,g
no,g
0.3
0.2
0.1
0
= 0.18
= 0.29
= 0.49
= 0.55
= 0.68
= 0.15
= 0.18
= 0.29
= 0.49
= 0.55
= 0.68
no,g
no,g
no,g
no,g
no,g
no,g
4000
3000
2000
1000
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p wf /p bar
Dimensionless IPR plot.
500
1000
1500
2000
Oil or Gas flowrate
Base IPR plot.
6000

p
2
p


qx 1  ν wf   (1  ν )  wf 
x
x 
4000


qx,max



p
p



3000
5000






(x = phase (e.g., oil, gas, water))
2000
 Condensate
or gas IPR parameter (gas condensate reservoirs)
1000
0 Low no or ng values — IPR more concave.
2000
 0Exact500value1000
of not1500
crucial
— similar curves for different no or ng values.
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 45
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 06:
Semi-Analytical Rate Relations
for Oil and Gas Flow
(End of Lecture)
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Semi-Analytical Rate Relations
for Oil and Gas Flow
Slide — 46